How to deform multiple mesh objects with one manipulator?
You should be able to do this with an armature.

Here a simple Rubik's cube like mesh object parented to an armature. 

The second cube is a linked duplicate.



The cube blocks on the face that rotate have been assigned to a vertex group, as we can verify in Weight Paint Mode.



After creating your armature make sure the bone is in the right place and has no rotation issues. You can lock the transforms so it doesn't rotate in undesired ways.



When parenting the cube to the armature Set Parent to "Armature Deform with Empty Groups".



Ensure that the groups and bones have matching names.



Here's the .blend from the screen shots in case I forgot to explain anything: